The job of the mixer is to ensure the music sounds good in as many speakers as possible, not just well designed speakers. I believe this is the theory behind using "fatiguing" speakers for mixing where you would never use it for luxury listening but need to use it to "hear into" worst parts of the
I agree with this.
I guess these days room correction is widespread in mixing, which of course further takes "character" out of the main monitor's FR.
I think best results are achieved by mixing on a good system (nice FR) then checking on non-ideal systems. Which is, so far as has been possible at any point in history, what mixers have always done.
Personally my approach is (if there's time!) once the mix is "done" on the main monitors, listen for problems on worse speakers and see if there's anything that stands out. If it does, go back to the mains and listen for the same thing. It might be that it was borderline and didn't get addressed on the mains (e.g. a vocal that's a bit sibliant in one overdub, or whatever....) but having heard the issue on a speaker that highlights that region, I go "oh actually, yes, that could be tweaked". Sometimes I go back to the mains and decide that what I heard on the alt speakers was just a function of their imperfections, so I don't adjust the mix for that.
For me it's often errors in matching of different recordings that stand out on different speakers. For example where a studio overdub has been patched in to a live performance, or an ADR recording to a production one. Sometimes though it's masking issues, such as; can I hear something clearly enough or is it masked by other content in the mix.
Although I don't deliberately do so myself, I do sometimes wonder if there's merit in mixing with more midrange, such as the upper-mid pushes we see on some high end studio monitors across various frequencies. By that I mostly mean that while it might not create the perfect spectral balance overall, it might show up mismatch and confict issues often exist in such frequencies. Dunno...